Research On Non-linear Question Of Dynamics In Biological System With Impulse Perturbations

Addressed in this dissertation is the dynamic complexity of biological system withimpulse perturbation. Several relevant problems attracting more and more attention in thisfield are discussed in detail, and a series of well-established, systematical, and importantresults are obtained.The backgrounds and research status on non-linear question of complex dynamic inbiological system are introduced. It also provides the readers with some preliminaries andimportant definitions and lemmas that are frequently used in the remaining chapters.On the basis of the mathematical theories and biological significance, a series ofbiological systems with impulse perturbation and interspecific competition are studiednumerically and analytically. Mathematical theoretical works have been pursuing the in-vestigation of the existence, locally asymptotically stable and globally asymptotically sta-ble for the semi-trivial periodic solution and permanence. Numerical analysis indicatesthat the key factors for long-term persistence are interspecific competition and impulseperturbation. Chaos spontaneously appears when the release amount is larger than somecritical value. We also show that although the impulse perturbation may be critical forcomplex dynamics, it cannot prevent the indirect effect on complex population dynamicscaused by interspecific competition. In addition, the largest Lyapunov exponent is com-puted. This computation demonstrates the chaotic dynamic behavior of the biologicalsystem. The qualitative nature of concerned strange attractors is also investigated throughtheir computed Fourier spectra, this foregoing investigation has the potential to be usefulfor the study of indirect effect due to noise. Furthermore, the consequence for conserva-tion management is that choosing the value of some critical parameters can in fact profitthan suffer from declining the amount of competitor population.Biological systems with impulsive control strategy and distributed time delay aredeveloped using the theory and methods of ecology and ordinary differential equations.Using the theory of impulsive equations, small amplitude perturbation skills, and compar-ison techniques, these systems are shown to be permanent and have a semi-trivial periodicsolution which is globally asymptotically stable. Further, the in?uences of the impulsiveperturbation and distributed time delay on complex dynamics are studied numerically and are found to depict rich dynamics. In addition, the largest Lyapunov exponent is com-puted. This computation demonstrates the chaotic dynamic behavior of the biologicalsystem. The qualitative nature of concerned strange attractors is how to be affected bydistributed time delay by computing their Fourier spectra. Finally, it is hope that theseinvestigations have the potential to be useful for the study of dynamical behaviors of bio-logical system with distributed time delay.On the basis of the theories and methods of ecology, the theory of impulsive equa-tions, small amplitude perturbation skills, and comparison techniques, conditions whichguarantee the existence and the global asymptotical stability of the prey-eradication pe-riodic solution are obtained. Further, the in?uences of the impulsive perturbation andseasonal effects on the inherent oscillation are studied numerically. These show to be con-sistent with the theoretical analysis and rich complex population dynamics, such as chaos,species extinction and permanence. Moreover, the population dynamical behavior of thesystem is demonstrated by computing the largest Lyapunov exponent. By investigatingthe strange attractors through their computed Fourier spectra, we know that seasonalityhas a profound effect on the population dynamical behavior.In a word, all these results are expected to be of use in the study of dynamic com-plexity of ecosystems.